Find the complex number $z$ such that $z^{2} + | z | = 0$ .

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Solution

$\Rightarrow z^{2} = - | z |$

Apply Mod both sides.

$\Rightarrow z^{2} = | - | z | | = | z | (∵ | z | \geq 0)$

$\Rightarrow | z | (| z | - 1) = 0$

$\Rightarrow | z | = 0$ or $1$

Case $(1)$

$| z | = 0$

$\Rightarrow z^{2} + 0 = 0 \Rightarrow z = 0$

Case $(2)$

$| z | = 1$

$\Rightarrow z^{2} + 1 = 0$

$\Rightarrow z^{2} = - 1$

$\Rightarrow z = i, - i$

$∴ z = 0, i, - i$

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